Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.
Georg CantorRelated topics
bestow bound certain consistent determinacy development existent free given hand instance introduction mathematics number numbers particular real relationship respect stand circumstances conditionsRelated quotes
Gravitational force is simple, and a thing by itself, as also are electric and magnetic forces as long as the electric and magnetic poles stand at rest. But as soon as motion comes into the picture, the whole situation is changed. Forces of new kinds come into play, for moving electric charges exert magnetic forces in addition to the electric forces they exert when at rest, while moving magnets exert electric forces in addition to the magnetic forces they exert while at rest. When the exact laws governing these intricate laws had been discovered by a great number of experimenters, Clerk Maxwell succeeded in expressing them in a mathematical form which was both simple and elegant.
James Jeans
Underlying the concept of positivity is the conviction that the positive is intrinsically positive in itself, without anyone pausing to ask what is to be regarded as positive. ... It is significant and really quite interesting that the term 'positive' actually contains this ambivalence. On the one hand, 'positive' means what is given, is postulated, is there-as when we speak of positivism as the philosophy that sticks to the facts. But, equally, 'positive' also refers to the good, the approvable, in a certain sense, the ideal. And I imagine that this semantic constellation expresses with precision what countless people actually feel to be the case.
Theodor Adorno
Let us get down to fundamentals. Is this an open, or is this a closed society? Is it a society where men can preach ideas - novel, unorthodox, heresies, to established churches and established governments - where there is a constant contest for men's hearts and minds on the basis of what is right, of what is just, of what is in the national interests, or is it a closed society where the mass media - the newspapers, the journals, publications, TV, radio - either bound by sound or by sight, or both sound and sight, men's minds are fed with a constant drone of sycophantic support for a particular orthodox political philosophy? I am talking of the principle of the open society, the open debate, ideas, not intimidation, persuasion not coercion...
Lee Kuan Yew
In its conception the literature prize belongs to days when a writer could still be thought of as, by virtue of his or her occupation, a sage, someone with no institutional affiliations who could offer an authoritative word on our times as well as on our moral life. (It has always struck me as strange, by the way, that Alfred Nobel did not institute a philosophy prize, or for that matter that he instituted a physics prize but not a mathematics prize, to say nothing of a music prize - music is, after all, more universal than literature, which is bound to a particular language.) The idea of writer as sage is pretty much dead today. I would certainly feel very uncomfortable in the role.
J. M. Coetzee
A second conspicuous landmark... is the enunciation of the fundamental law of radioactive disintegration by Rutherford and Soddy in 1903. This law was in no sense a consequence or development of Planck's theories; indeed fourteen years were to elapse before any connection was noticed between the two. The new law asserted that the atoms of radioactive substances broke up spontaneously, and not because of any particular conditions or special happenings. This seemed to involve an even more startling break with classical theory than the new laws of Planck; radioactive break-up appeared to be an effect without a cause, and suggested that the ultimate laws of nature were not even causal.
James Jeans
